Time-Dependent Determinantal Point Processes

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The proposal focuses on the study of time-dependent discrete and continuous probabilistic models of random matrix type. More exactly, we work with time-dependent determinantal point processes. We use three different approaches corrersponding to three sources of our models: Interacting Particle Systems (the totally asymmetric exclusion process, the polynuclear growth process, etc.), Representation Theory of Big Groups (the infinite symmetric group, the infinite-dimensional unitary group and the like), and Schur processes, which may be viewed as a very general algebraic formalism suitable for various problems of Enumerative Combinatorics.Despite different origins these themes are closely interrelated, and they conveniently provide three different points of view on the class of time-dependent determinantal point processes.The proposal focuses on the study of large time fluctuations of various models of random growth in one space dimension. Such models naturally appear in mathematics and physics, and until recently no tools were available for computing their fine large time asymptotics.A few years ago a combination of methods from statistical physics, combinatorics, analysis, and representation theory lead to a stunning success: For a handful of models not only we were able to rigorously control the size of the fluctuations but we also found explicit distributions that play the role of the famous bell-shaped curve. We plan to turn this success into a method that delivers results for a much wider class of growth models while broadening the connection of the subject to other domains of mathematics and mathematical physics.